$$ {\displaylines\int\tan^4\left(5x\right)\differentialD x\\ \placeholder{}} $$

To solve the integral $ \int \tan^4(5x) \, dx $, we can rewrite $ \tan^4(5x) $ using trigonometric identities to simplify the expression.

### Step 1: Express $ \tan^4(5x) $ in terms of $ \tan^2(5x) $
We use the identity:
\[
\tan^2(\theta) = \sec^2(\theta) - 1
\]
Thus, $ \tan^4(5x) $ can be written as:
\[
\tan^4(5x) = (\tan^2(5x))^2 = \left(\sec^2(5x) - 1\right)^2
\]
Expanding the square:
\[
\tan^4(5x) = \sec^4(5x) - 2\sec^2(5x) + 1
\]

### Step 2: Split the integral
Now, we can split the integral into three simpler integrals:
\[
\int \tan^4(5x) \, dx = \int \left( \sec^4(5x) - 2\sec^2(5x) + 1 \right) dx
\]
This becomes:
\[
\int \sec^4(5x) \, dx - 2\int \sec^2(5x) \, dx + \int 1 \, dx
\]

### Step 3: Solve each integral
1. **Integral of $ \sec^4(5x) $:**

   Use the reduction formula for powers of secant:
   \[
   \int \sec^4(5x) \, dx = \frac{1}{3} \tan(5x) \sec^2(5x) + \frac{2}{15} \tan^3(5x) + C_1
   \]

2. **Integral of $ \sec^2(5x) $:**

   This is a standard integral:
   \[
   \int \sec^2(5x) \, dx = \frac{1}{5} \tan(5x)
   \]

3. **Integral of 1:**
   \[
   \int 1 \, dx = x
   \]

### Step 4: Combine the results
Putting everything together:
\[
\int \tan^4(5x) \, dx = \frac{1}{3} \tan(5x) \sec^2(5x) + \frac{2}{15} \tan^3(5x) - \frac{2}{5} \tan(5x) + x + C
\]
where $ C $ is the constant of integration.

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Would you like more details on any step?

Here are 5 related questions you might find interesting:
1. How would you solve $ \int \sec^n(x) \, dx $ for general $ n $?
2. Can you derive the reduction formula for $ \sec^n(x) $?
3. What is the integral of $ \tan^n(x) $ for general $ n $?
4. How do trigonometric identities simplify complex integrals?
5. How would you approach the integral of $ \cot^4(x) $?

**Tip:** Always look for identities to simplify trigonometric integrals before starting to integrate!

Solve the integral ∫ tan⁴(5x) dx using trigonometric identities and reduction formulas. Learn the detailed steps, including splitting the integral and applying standard formulas. Suitable for students studying undergraduate calculus and trigonometry.

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